Abstract

Let R be a commutative ring with 1, and M is a (left) R–module. We introduce the concepts of (strongly) pcompactly packed submodules as: A proper submodule N of an R-module M is called P-Compactly Packed if for each family { α}α ∈Λ N of primary submodules of M with α α N N ∈Λ ⊆ ∪ , there exists ∈ Λ n α ,α ,...,α 1 2 such that i N N ni =1 α ⊆ ∪ . If β N ⊆ N for someβ ∈ Λ , then N is called Strongly P-Compactly Packed. In this paper, we list some basic properties of this concept. In addition, the necessary and sufficient conditions for an R−module M to be (strongly) P-Compactly Packed are investigated.We also generalize the Prime Avoidance Theorem for modules that was proved in [7] to the Primary Avoidance Theorem for modules. Furthermore, we find the conditions on an R-module M that make the following important result true, that is for a multiplication Bezout module M, M is strongly P- compactly packed if and only if every primary submodule of M is strongly P- compactly packed.